Let's analyze and solve the problem step by step.
### Problem Statement
A flask can withstand a maximum pressure of 1.6 atm. It is initially filled with chlorine gas at 10°C and 764 mmHg. We need to determine the temperature at which the flask will break (explode) when heated.
### Given Data
1. Maximum pressure the flask can bear: [tex]\( P_2 = 1.6 \text{ atm} \)[/tex]
2. Initial temperature: [tex]\( T_1 = 10^\circ \text{C} \)[/tex]
3. Initial pressure: [tex]\( P_1 = 764 \text{ mmHg} \)[/tex]
### Solution Steps
#### Step 1: Convert the initial pressure to atmospheres
Since 1 atmosphere (atm) is equivalent to 760 mmHg, we convert the initial pressure [tex]\( P_1 \)[/tex] from mmHg to atm:
[tex]\[ P_1 = \frac{764 \text{ mmHg}}{760 \text{ mmHg/atm}} = 1.0052631578947369 \text{ atm} \][/tex]
#### Step 2: Convert the initial temperature to Kelvin
To convert Celsius to Kelvin, we use the formula:
[tex]\[ T_1 = 10^\circ \text{C} + 273.15 = 283.15 \text{ K} \][/tex]
#### Step 3: Use the ideal gas law relationship for constant volume
For an ideal gas under constant volume, the relationship between pressure and temperature is given by:
[tex]\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \][/tex]
We need to solve for the final temperature [tex]\( T_2 \)[/tex] in Kelvin:
[tex]\[ T_2 = \frac{P_2 \cdot T_1}{P_1} \][/tex]
Substituting the known values:
[tex]\[ T_2 = \frac{1.6 \text{ atm} \cdot 283.15 \text{ K}}{1.0052631578947369 \text{ atm}} = 450.66806282722507 \text{ K} \][/tex]
#### Step 4: Convert the final temperature back to Celsius
Convert [tex]\( T_2 \)[/tex] from Kelvin to Celsius:
[tex]\[ T_2 = 450.66806282722507 \text{ K} - 273.15 = 177.5180628272251^\circ \text{C} \][/tex]
### Conclusion
The flask will explode when the temperature reaches approximately [tex]\( 177.52^\circ \text{C} \)[/tex].
